Risk averse variants of Dynamic Programming are widely studied. Our work is very close to [1]} who, for a very similar setting, proves the convergence of the Value Iteration algorithm. The contribution of our work is two-fold. First, instead of complicated axiomatic definition, we defined the one-stage risk mapping constructively by means of its risk envelope, which is moreover independent of a choice of an underlying probability measure; consequently, the whole exposition, including proofs, is much simpler. Second, instead of the convergence of a particular solution algorithm, we prove the contractive property of the Bellman operator, which can be then plugged into convergence proofs of many different algorithms.
